Explore models for dynamical systems in one and two dimensions in this 57-minute lecture by André de Carvalho from the Universidade de São Paulo. Delve into the Milnor-Thurston theorem, which demonstrates that multimodal endomorphisms of the interval are semi-conjugate to piecewise linear maps with constant absolute slope (plcas), sharing the same topological entropy. Examine measurable pseudo-Anosov surface homeomorphisms as potential 2-dimensional analogs to plcas interval endomorphisms. Consider a conjectural extension of the Milnor-Thurston theorem for sufficiently smooth surface diffeomorphisms. Discover connections between this topic and Teichmüller Theory, as well as the geometry and topology of 3-manifolds. This lecture is part of the Simons Semester on Dynamics series, offering advanced insights into dynamical systems theory.